To write an equation whose graph will pass through the point [tex]\((2, 3)\)[/tex], we can use the general form of a linear equation in slope-intercept form, which is:
[tex]\[ y = mx + b \][/tex]
Here, [tex]\(m\)[/tex] represents the slope of the line, and [tex]\(b\)[/tex] is the y-intercept.
Step-by-Step Solution:
1. Identify the Known Point:
We are given a point [tex]\((2, 3)\)[/tex], meaning when [tex]\(x = 2\)[/tex], [tex]\(y\)[/tex] should be [tex]\(3\)[/tex].
2. Assume a Value for the Slope [tex]\(m\)[/tex]:
Since there are infinitely many lines that can pass through the point [tex]\((2, 3)\)[/tex], we can choose any value for the slope [tex]\(m\)[/tex]. For simplicity, let's assume [tex]\(m = 1\)[/tex].
3. Substitute the Point into the Equation:
Using the slope [tex]\(m = 1\)[/tex] and the point [tex]\((2, 3)\)[/tex],
[tex]\[ 3 = 1 \cdot 2 + b \][/tex]
4. Solve for the Y-Intercept [tex]\(b\)[/tex]:
[tex]\[ 3 = 2 + b \][/tex]
[tex]\[ b = 3 - 2 \][/tex]
[tex]\[ b = 1 \][/tex]
5. Form the Equation:
Now that we have the slope [tex]\(m = 1\)[/tex] and the y-intercept [tex]\(b = 1\)[/tex], we can write the equation of the line:
[tex]\[ y = 1x + 1 \][/tex]
Simplified, this equation is:
[tex]\[ y = x + 1 \][/tex]
Thus, one possible equation of a line that passes through the point [tex]\((2, 3)\)[/tex] is:
[tex]\[ y = x + 1 \][/tex]
Infinite Possible Equations:
There are infinitely many such equations, as the slope [tex]\(m\)[/tex] can be any real number. By changing the slope [tex]\(m\)[/tex], while still satisfying the condition [tex]\( y = mx + b \)[/tex] where it passes through [tex]\((2, 3)\)[/tex], you can generate endless possible equations.
Summary:
One possible equation of a line passing through the point [tex]\((2, 3)\)[/tex] is:
[tex]\[ y = x + 1 \][/tex]
However, since the slope [tex]\(m\)[/tex] can take any value, there are infinitely many such lines that can pass through the point [tex]\((2, 3)\)[/tex]. The number of such equations is infinite.
